An approach to the Weinstein conjecture via J-holomorphic curves

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Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

An approach to the Weinstein conjecture

via J-holomorphic curves

Candidato:

Gabriele Benedetti

Relatore:

Prof. Alberto Abbondandolo

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Introduction

The aim of this work is to give a concise introduction to the Weinstein conjecture and to analyze a proof of the conjecture in a particular case. The Weinstein conjecture is a meeting point between two important elds of mathematics: dynamical systems and contact-symplectic geometry. On a closed compact odd-dimensional manifold Σ, endowed with a contact 1-form α, it is well-dened a nowhere vanishing vector eld Rα, called the Reeb

vector eld of α. The Weinstein conjecture claims that Rα has a periodic

solution.

As regard the dynamical point of view, when Σ = S3 _{we can interpret}

this statement as a particular case of the Seifert conjecture, which asserts that every nowhere vanishing smooth vector eld on S3 _{has a periodic orbit.}

The Seifert conjecture was disproven in 1994 by K. Kuperberg, who showed that nowhere vanishing vector elds without periodic orbits do exist on any compact closed odd-dimensional manifold.

If we restrict the class of vector eld a little more, entering in the realm of symplectic geometry, we come to the Hamiltonian vector eld. Suppose that Σ can be embedded in a symplectic manifold (M, ω) and that there exists a function H : M → R, such that Σ = H−1₍₀₎ _{(i.e. Σ is the 0 energy}

level) and 0 is a regular value for H. Then the Hamiltonian vector eld XH

on M associated to H, resticts to a nowhere vanishing vector eld on Σ. The corresponding existence conjecture for vector elds of this kind is called the Hamiltonian Seifert conjecture and was disproved in 1999 by Herman when the dimension of M is strictly bigger than 4.

On the other hand, positive results under additional hypotheses were known from the end of the Seventies. In 1978 Alan Weinstein proved that if the energy level Σ is the boundary of a convex domain in R2n_{, then it}

carries periodic orbits. In the same year Rabinowitz generalized this the-orem proving that it is sucient to suppose that Σ is the boundary of a star-shaped domain. These achievements deeply impressed the mathemati-cians who worked on Hamiltonian dynamics. Many thought that these theo-rems could prelude to further developments. However, as Weinstein himself pointed out, the hypotheses used to prove the existence of periodic orbits

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were not satisfactory. The dynamics of a Hamiltonian system in a symplectic manifold is invariant under dieomorphisms which preserve the symplectic structure, hence the notion of having a periodic orbit is invariant under the action of this group. On the contrary both the convexity and the star-like assumptions are not invariant. Weinstein introduced in 1979 a property that on the one hand could generalized the star-shaped hypothesis and on the other hand were well-dened in an abstract symplectic context. This is the notion of hypersurfaces of contact type, that allowed Weinstein to state his famous conjecture:

(Original Weinstein Conjecture). Let (M, ω) be a symplectic manifold and H : M → R a smooth function. Suppose that 0 is regular value for H and Σ := H−1₍₀₎ _{a hypersurface of contact type, with H}1_{(Σ, R) = 0. Then}

the Hamiltonian eld on Σ carries a periodic orbit.

Nowadays the homological hypothesis has been abandoned since reputed unnecessary and the problem has been reformulated within a genuine con-tact geometric framework in the following way:

(Weinstein Conjecture). Let Σ be a compact closed manifold endowed with a contact form α. manifold. The Reeb vector eld of α carries a periodic orbit.

The conjecture in this generality is still open. In this thesis we are going to prove only a particular case.

Main Theorem. Every compact hypersurface, which is restricted contact type and displaceable in an exact and convex at innity symplectic manifold carries a closed Reeb orbit.

For the convenience of the reader we include here a short summary of the content of each chapter.

In the rst chapter we give an introduction to basic notions in contact and symplectic geometry and describe some concrete and important exam-ples, where the conjecture is mainly studied: Stein manifold and the particle in a magnetic eld are the two most relevant instances.

In Chapter 2, we give an account of the approaches to the proof, which have been developed so far. In particular we dwell on methods based on a theorem of existence on almost every energy level, due to Hofer and Zehnder. The interest to this technique relies on the fact that the hypotheses at the ground can be compared to those of the Main THeorem introduced above.

The proof of the Main Theorem itself is developed from Chapter 3 to 6. In chapter 3 we dene A, the Hamiltonian action functional on E0, the

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by Rabinowitz in the proof of his already mentioned theorem. It is interesting for the following reason: its nontrivial critical points are the periodic orbits we seek. In order to study the critical set we consider the space M, composed by paths w from R to E0, solving a gradient-like equation

dw

ds = −∇A(w), (∗)

and satisfying particular boundedness conditions for the derivative and with a prescribed behaviour at innity. Expliciting (∗) we nd that it is an order 0perturbation of the equation of J-holomorphic curves from the cylinder T × R in M :

∂su + Ju∂tu = 0,

where J is an almost complex structure on M, compatible with ω. This partial derivative equation has been studied for the rst time in 1985 by Gromov and its properties are essential throughout the proof.

In the fourth chapter we endow M with the C∞

loc-topology and show that

the topological space we get is sequentially relatively compact. The calculations needed to arrive to this result are a generalization of those used by Cieliebak and Frauenfelder in 2009 for the denition of the Rabinowitz Floer Homology of an hypersurface. However, our proof is direct and does not require the construction of such homology, which relies on cumbersome transversality arguments.

In Chapter 5 we investigate the asymptotic propertis of elements in M. Morse-Bott theory turns out to be applicable in this case.

Finally in chapter 6 we use Fredholm Theory to show that M is not a C_loc∞-closed space. Putting together the results from the preceding chapters, we arrive to the existence of a limit pointwbnot belonging to M. Analyzing the behavior at innity of the function wb, we succeed in nding a periodic orbit and thus in proving the Main Theorem.

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Chapter 1

Preliminaries

This chapter aims to construct the language and the environment needed to understand the conjecture in its full generality. Therefore we begin with an introduction to the basic denitions and guiding examples from symplectic and contact geometry.

1.1 An introduction to symplectic geometry

The Hamiltonian formulation of the dynamics' problem was a fruitful approach in the study of classic physical systems. It is enough to mention here KAM theory (`50 -`60) which has become the cornerstone of the theory of perturbation. Symplectic geometry was born to give a coordinate-free description of the Hamilton equation when the phase space is an abstract manifold and not only a domain in an Euclidean space.

Within this chapter all the objects belong to the smooth category. Denition 1.1.1. A symplectic manifold is a couple (M, ω) where M is a manifold and ω is a closed 2-form on M which is nondegenerate, i.e. the following implication holds ∀z ∈ M:

∃v ∈ TzM, ∀u ∈ TzM ωz(u, v) = 0 ⇒ v = 0 .

In the following discussion we use the notations:

If V is a subbundle of T M then Vω _{is the subbundle whose bers are}

dened by

(Vω)z:= {u ∈ TzM | ∀v ∈ Vz, ωz(u, v) = 0} .

If v ∈ T M and η is a k-form on M, then ιvη := η(v, ·)

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Then the nondegeneracy condition can be written more concisely as (T M )ω = 0 or ιvω = 0 ⇒ v = 0

and it establishes the following linear isomorphism T M → T∗M

v 7→ ι_vω.

Remark 1.1.2. We have dened the form ω by two properties.

a) The nondegeneracy is a punctual property. It is a condition for ωz as a

bilinear antisymmetric form on TzM and can be generalised to arbitrary

vector bundles.

We call (E, ω) a symplectic vector bundle if E → M is a vector bundle over a manifold and ω : E × E → R is a bilinear nondegenerate antisymmetric form on each ber. Since ω is nondegenerate the rank of E is even. Indeed, suppose E 6= 0 and x a point z ∈ M. The dimension of Ez can't be one because every antisymmetric form on R is zero. So we can

pick in Ez two linearly independent vectors u1, v1such that ω(u1, v1) = 1.

Then the nondegeneracy yields

Ez = Span(u1, v1) ⊕ Span(u1, v1)ω

and ω restricted to both this subspaces is nondegenerate. Now the con-clusion follows from induction. In this way we get as a byproduct a basis for Ezmade by vectors (u1, v1, . . . , un, vn)such that, if (u1, v1, . . . , un, vn)

is the dual basis, we can write

ωz = n

X

k=1

uk∧ vk.

Since we can perform this construction smoothly in a neighbourhood of z we have found canonical local frames in which the symplectic vector bundle has a simple model.

From this model we see that a symplectic vector bundle is orientable (and so the same is true for a symplectic manifold).

In fact ω ∧ ω ∧ . . . ∧ ω

| {z }

ntimes

is a volume form on E. Its expression using coor-dinates induced from a local frame is

n! u1∧ v1∧ · · · ∧ un∧ vn , which is nowhere vanishing.

b) The closedness of ω is a local property. It describes how the forms on each ber t together and it is responsible for the existence of canonical local coordinates. Namely it is possible to choose the frames described above as coordinate vectors frames. This is the content of Darboux's Theorem.

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Theorem 1.1.3 (Darboux). Let (M, ω) be a symplectic manifold and z ∈ M. Then there exists coordinates (p1_{, q}1_{, . . . , p}n_{, q}n₎ _{in a neighbourhood U of z}

such that ω|U = n X k=1 dpk∧ dqk.

Darboux's Theorem says that there is a unique local model for symplectic manifold. So now we will take a closer look to this standard structure. Example 1.1.4. Consider Cn_{as a complex vector space. The moltiplication}

by a scalar is made componentwise. Let us denote by J the moltiplication by the imaginary unit: it is a C-linear automorphism of Cn _{such that J}2 _{= −1}_.

Consider now the n standard coordinate vectors ∂zk and their dual basis dzk

so that a vector can be written as u = Pkdzk(u)∂zk. Then dene

dpk := <(dzk) and dqk:= =(dzk) and an R-linear isomorphism with R2n _{as follows:}

u 7→ (dp1(u), dq1(u), · · · , dpn(u), dqn(u)). If we set

∂_pk := ∂_zk and ∂_qk := J ∂_zk ,

then this isomorphism gives the coordinates of a vector in Cn _{with respect}

to this R-basis. From now on we always consider Cn _{as a real vector space}

equipped with an endomorphism J that acts on it as follows: J ∂_pk = ∂_qk J ∂_qk = −∂_pk.

Consider the following two additional structure on Cn_.

1. Euclidean: a real scalar product g(u, v) =X k dpk(u)dpk(v) + dqk(u)dqk(v) .

2. Symplectic: a bilinear antisymmetric form ω(u, v) =X

k

dpk∧ dqk(u, v).

The complex structure relates these bilinear forms by the formula g(u, v) = ω(J u, v).

So we only need two among g, ωandJ in order to nd the last one.

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take an open set V ⊂ Cn_{, regarded as a real manifold, and use the canonical}

isomorphism between TzV and Cnin order to transfer the above structures on

T V (here we mean the real tangent space). If zkare the complex coordinates and pk _{:= <(z}k₎_{, q}k _{:= =(z}k₎ _{are the real coordinates then the notations}

used above for vectors and forms ts with the usual meaning those symbols have in dierential geometry, for example dpk_{indicates the dierential of the}

real function pk_.

The form ω we obtain becomes a symplectic form on T V . Indeed, since w is constant, dω = 0.

In this case we have found that the symplectic form and the complex structure on V are compatible in some sense. This can be generalized as follows.

Denition 1.1.5. Let (M, ω) be a symplectic manifold and J : T M → T M an almost complex structure, i.e. J is a bundle map such that J2 ₌

− id_{T M}. J is said to be compatible with ω if

gz(u, v) := ωz(Jzu, v), u, v ∈ TzM

is a metric on M (in other words (M, g) becomes a Riemannian manifold). For every xed symplectic manifold (M, ω) the set

Jω:= {J is compatible with ω}

is nonempty and contractible, so T M is a well-dened complex vector bundle (see (34) for further details). Every complex manifold M carries a natural almost complex structure (and if a map J arises in this way is said inte-grable), however if M is also symplectic, this does not imply that the two structures are compatible in the sense given above. If this turns out to be the case M is called a Kähler manifold. A distinguished class of Kähler manifolds is described in the next example.

Example 1.1.6 (Stein manifolds). Let V be a complex open manifold and let J be the associated integrable structure on T V . A function f : V → R is exhausting if it is proper and bounded from below and is strictly plurisub-harmonic if the exact 2-form ω = d (df ◦ J) is such that

ωz(Jzv, v) > 0, ∀v ∈ TzV, v 6= 0.

If V admits an exhausting strictly plurisubharmonic function f then it is called a Stein manifold and we will write (V, J, f) to denote it.

Observe that the above inequality implies that ω is nondegenerate and, since it is also exact, it is actually a symplectic form and hence V is a symplectic manifold. Since J is integrable ω is of type (1, 1) with respect to the splitting of TCV induced by J. Then J is ω-compatible since

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In Cn _{the function z 7→ |z|}2 _{is an exhausting plurisubharmonic function.}

Indeed

d d|z|2◦ J = d (2pdp)◦ J + (2qdq)◦ J = 2d pdq − qdp = 4dp ∧ dq. Therefore up to a constant factor we get the standard symplectic form.

Let us continue now with an example from classical physics.

Example 1.1.7 (Cotangent bundles). Let M be a smooth manifold and π: T∗M → M the cotangent vector bundle. We dene a 1-form λ on T (T∗M ) as follows:

∀η ∈ T∗M, ∀v ∈ Tη(T∗M ) , λη(v) = ηπ(η)(dηπ(v)).

λis characterised by the following property:

∀η : M → T∗M, η∗(λ) = η.

Then (T∗_{M, dλ)} _{is a symplectic manifold. Indeed dλ is a closed form and}

if we choose coordinates (pk_{, q}k₎ _{on T}∗_M _{that are induced from coordinates}

(qk₎_{on M then we nd that locally λ = P}

kpkdqk. Its dierential is locally

P

kdpk∧ dqk, which we have seen to be nondegenerate.

This class of examples encloses also the case of Cn _{because R}2n∼_{= T}∗

Rn. In the previous examples the symplectic form was actually exact. This additional property will be relevant in what follows and so we include it in a denition.

Denition 1.1.8. A symplectic manifold (M, ω) is said to be exact if exists a 1-form λ on T M (called a Liouville form), such that ω = dλ. Since often the 1-form itself is more important than its symplectic dierential we shall denote an exact manifold by (M, λ) rather than (M, dλ).

Remark 1.1.9. The exactness of ω implies the exactness of ωn_{, which is}

a volume form on M. This fact implies that an exact manifold can't be closed. For the same reason if we rotate the perspective, a closed manifold with H2

dR = 0 cannot carry any symplectic structure.

We dene now the dieomorphisms and the vector elds compatible with the symplectic structure.

Denition 1.1.10. A dieomorphism F : M → M0 _{between (M, ω) and}

(M0, ω0) is a symplectomorphism (or is symplectic) if F∗ω0 = ω.

A vector eld X on (M, ω) is a symplectic vector eld if L_Xω = 0.

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Remark 1.1.11. We make the following two observations regarding this denition.

Darboux's Theorem is equivalent to saying that locally every symplec-tic manifold is symplectomorphic to an open set in Cn _{with the}

stan-dard symplectic structure. So from a local point of view all symplectic manifolds look the same.

Vector elds can be seen as the innitesimal counterpart of dieomor-phism. For every real t we can consider Φtthe ow at time t associated

to X. This is a dieomorphism between two open sets in M (possibly empty) and is symplectic if and only if X is symplectic too. Indeed, if t ≥ 0and z is a point in the domain of Φt, then is in the domain of Φs

for 0 ≤ s ≤ t, too. Since Φ0 = Id, Φ0 is obviously symplectic. So,

∀t (Φ∗_tω)_z= ωz ⇐⇒ _dtd (Φ∗tω)z = 0 ⇐⇒ Φ∗_t (LXω)Φt(z) = 0 ⇐⇒ L_Xω = 0.

Moreover Cartan's formula yields:

LXω = ιXdω + d(ιXω) = d(ιXω) .

This allows us to rewrite the condition of being symplectic: L_Xω = 0 ⇐⇒ d (ιXω) = 0 .

At the beginning of this section we have pointed out that ω establishes an isomorphism between vector elds and 1-forms. Therefore if we want to construct a symplectic vector eld we only need to pick a closed form η and then get X from the equality η = ιXω.

The easiest closed forms are the dierentials of functions on M. This will give the vector elds which we are interested in.

Denition 1.1.12. Let (M, ω) be a symplectic manifold and H : M → R a function on it. We call the vector eld XH dened by

ιXHω = −dH

an Hamiltonian vector eld and H the Hamiltonian of the system. Then the equation

˙

z = XH(z). (1.1)

represents the Hamiltonian formulation of the problem of dynamics for a classical physical system.

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The function H can be viewed as the energy of the system and it is preserved during the motion. Indeed, if z is a trajectory, then

dH

dt (z(t)) = dH (XH(z(t))) = −ω (XH(z(t)), XH(z(t))) = 0.

In this sense we say that autonomous Hamiltonian systems are conservative. The study of Equation 1.1 can be carried out along dierent lines de-pending on what is the goal one person has in mind. For example one may be concerned with quantitative estimates as well as stability issues or topo-logical properties of trajectories. The focus of our enquiry will be on the last class of problems. In particular we shall investigate which general hypotheses can be imposed in order to guarantee

the existence of periodic solutions for the ordinary dierential equation (1.1) associated to an Hamiltonian H

in a given energy level.

However before starting with an analysis of the problem from an abstract point of view we will dwell a little more on the connection between symplectic geometry and physics.

1.2 From Newton's law to the Hamilton equations

Consider a particle (or a physical system) that moves in a Riemannian manifold (M, g) under the action of a force f, where f : T M → T M is an arbitrary function. If we set ∇ for the Levi-Civita connection on π : T M → M induced by g, then an admissible trajectory γ : (a, b) → M satises the Newton's law:

∇γ˙˙γ = f ( ˙γ), (1.2)

where ∇γ˙ is the covariant derivative for vector elds along γ. This is a second

order dierential equation for curves on M, but we can nd an equivalent rst order equation for its velocity ˙γ.

On T (T M) is canonically dened the vertical subbundle V whose ber at v ∈ TqM is the image of the injective linear maps

Iv: TqM → Tv(T M ) u 7→ d dt _t=0(v + tu).

Equivalently V is the kernel of the bundle map dπ : T (T M) → T M.

Moreover the connection gives rise to a subbundle H of T (T M) which is called the horizontal subbundle and which is a direct summand of V, i.e. T (T M ) = V ⊕ H. It can be dened through the injective maps

Lv: TqM → Tv(T M )

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where ˜v : M → T M is an arbitrary extension of v to a vector eld on M. Then

Hv:= Lv(TqM )

and Lv is a right inverse for dvπ, namely

dvπ ◦ Lv = idTqM. (1.3)

Set now v := ˙γ and apply Iγ˙ to both sides of Equation (1.2) obtaining the

equivalent equation

Ivf (v) = Iv(∇vv) = dv(v) − Lv(v), (1.4)

where we have substituted for ∇vv using the denition of Lv. Set

F (v) := Ivf (v)

and dene the geodesic vector eld G: T M → T (T M) as G(v) := Lv(v).

Then (1.4) can be rearranged into

˙v = G(v) + F (v). (1.5) Remark 1.2.1.

Observe that, since G is horizontal and F is vertical, the vector eld on the right hand side of (1.5) respects the splitting on T (T M) induced by g.

Furthermore if F = 0 the solutions are precisely the geodesics of (M, g), hence the adjective `geodesic' for G.

It is interesting to notice that g gives rise to the bundle isomorphisms T∗M → T M,] T M → T[ ∗M.

Then we can

endow T∗_M _{with the pullback metric k := ]}∗_g_,

obtain an equation for η := [v on T∗_M _{that is equivalent to (1.5),}

˙

η = d_(]η)[(G(]η) + F (]η)) = d_(]η)[(G(]η)) + d_(]η)[(F (]η)). (1.6) As is clear from (1.6) we can analyse the pushforward of F and G separately. For brevity we set

( _ˆ

G(η) := d_(]η)[(G(]η)), ˆ

F (η) := d(]η)[(F (]η)).

The crucial point is that these vector elds are indeed Hamiltonian with respect to the standard structure we dened in Example 1.1.7.

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Proposition 1.2.2. Let (T∗_{M, dλ)} _{be the standard symplectic structure on}

the cotangent bundle ˆπ : T∗_{M → M} _{of a Riemannian manifold (M, g).}

Using the previous notations we dene K(η) := 1

2kˆπ(η)(η, η) and we suppose

that f = −(∇V ) ◦ π, where V : M → R is a real function. Then ˆ

G = XK, F = Xˆ V◦ˆπ.

Setting H := K+V◦ˆπ, we see that (1.5) can be written as ˙

η = XH(η).

Remark 1.2.3. K represents the kinetic energy of the system. K is convex along the bers.

V ◦ ˆπ represents the potential energy. When f admits a potential physicists say that the force is conservative.

If we write down this equation using local coordinates (p, q), we recover the Hamilton equation of classical physics. The following identity holds:

ιXHdλ = ιXH(dp ∧ dq) = dp(XH) · dq − dq(XH) · dp. Furthermore, −dH = −∂H ∂p · dp − ∂H ∂q · dq.

From these equations we obtain the components of XH. Substituting in (1.1)

we get the familiar

       ˙ p = −∂H ∂q ˙ q = ∂H ∂p .

During the Eighties the case of a charged particle immersed in a magnetic eld became the subject of an intensive research. The Lagrangian (which we will not discuss here) and the Hamiltonian approach were carried out by Novikov and Tamainov, who used a generalization of Morse theory to multivalued functionals (37; 38), and by Arnol'd, who in addition exploited techniques from symplectic geometry (6). Their research was continued fur-ther by scholars such as V. Ginzburg (21; 22), G. Contreras (12) and G. P. Paternain (13). Since the Weinstein conjecture has been proven positively for systems belonging to this category, now we shall describe shortly what the problem is about.

Example 1.2.4 (Particle in a magnetic eld). In the three-dimensional Euclidean space the Maxwell equations for the magnetic eld B yield

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If we dene

σ := ιB(dq1∧ dq2∧ dq3),

then σ is a closed 2-form on R3_.

Moreover a particle which has a unitary charge is subject to the Lorentz force f ( ˙q) = ˙q ×B.

A simple calculation shows ˙

q ×B = ] (ιq˙σ) .

So we can generalize this situation to an arbitrary triple (M, g, σ), where (M, g) is a Riemannian manifold and σ is a closed 2-form on M. With the notation as above the corresponding vector eld ˆF on T∗M is given by

ˆ

F = ˆIη(ι]ησ),

where ˆI is the vertical lift from M to the cotangent bundle T∗_M_.

Further-more we can use σ to dene the twisted 2-form on T∗_M

ωσ:= dλ − ˆπ∗σ,

which is easily seen to be symplectic. The following proposition shows the connection between magnetic elds and symplectic geometry.

Proposition 1.2.5. Let (M, g, σ) be dened as before and consider a charged particle on M subjected to a force of the form

f (v) := −∇V (π(v)) + ] (ιvσ) . (1.7)

Then the corresponding Newton's equation is equivalent to a Hamilton equa-tion with respect to the twisted symplectic structure (T∗_{M, ω}

σ). The

Hamil-tonian of the system is H = K + V ◦ ˆπ.

Moreover if the magnetic eld is exact, i.e. σ = dα, the following trans-lation map is a symplectomorphism

Ψα: (T∗M, dλ) → (T∗M, ωdα)

η 7→ η + α_ˆ_π(η).

Therefore we get an equivalent Hamiltonian system on (T∗_{M, dλ)} _{with the}

Hamiltonian function obtained by substitution

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Proof. First notice that for every vertical vector eld X on T∗_M_{, we have}

(

ιX(ˆπ∗σ)_η = 0 ,

ιX(dλ)_η = πˆ∗( ˆIη−1(X)) .

(1.8)

Furthermore from Equation (1.3) we nd ι_Gˆπˆ∗σ η = (ˆπ ∗_σ) η( ˆG(η), ·) = σ_π(η)_ˆ (dηˆπ( ˆG(η)), dηπ(·))ˆ = σπ(η)ˆ (d(]η)π(G(]η)), dηπ(·))ˆ = σ_π(η)_ˆ (]η, dηπ(·))ˆ = ˆπ∗ ι(]η)σ η. We calculate now ι_{G+ ˆ}ˆ _Fωσ. ι_{G+ ˆ}ˆ _Fωσ = ι_Gˆωσ+ ι_Fˆωσ = ι_Gˆdλ − ι_Gˆπˆ∗σ + ι_Fˆdλ − ι_Fˆωσ = −dK − ι_Gˆπˆ∗σ + ι_Iˆ (·)(ι](·)σ)dλ − d(V ◦ ˆπ) = −dK − ˆπ∗ ι(]·)σ + ˆπ∗ ˆI_(·)−1Iˆ(·)(ι](·)σ) − d(V ◦ ˆπ) = −dK − d(V ◦ ˆπ)

Suppose now that σ = dα. First we nd that (Ψ∗_αλ)_η(ξ) = λΨα(η)(dηΨα(ξ))

= Ψα(η)(dΨα(η)πdˆ ηΨα(ξ))

= Ψα(η)(dηπ(ξ))ˆ

= η(dηπ(ξ)) + αˆ π(η)ˆ (dηπ(ξ))ˆ

= (λ + ˆπ∗α)_η(ξ). Using this identity we get

Ψ∗_α(ωdα) = Ψ∗α(dλ) − Ψ∗α(ˆπ∗σ)

= d (Ψ∗_αλ) − (Ψα◦ ˆπ)∗σ

= d(λ + ˆπ∗α) − ˆπ∗σ = dλ.

Remark 1.2.6. The rst part of the proposition indicates that the intro-duction of a magnetic term in the force aects the symplectic geometry of the cotangent bundle while the Hamiltonian function remains unchanged.

As regard the second part we nd the following byproduct: if α is closed, i.e. σ = 0, Ψα is a symplectomorphism from the standard symplectic

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1.3 The contact hypothesis

As we have said at the end of the previous section we are looking for periodic solutions of Equation (1.1). The rst task will be to describe the additional hypotheses Weinstein included in the formulation of his conjec-ture about the existence of a periodic orbit.

We have observed before that H is a constant of the motion. Thus we can focus our attention on a xed set Σc:= {H = c}, because it is invariant

under the ow of XH.

The rst hypothesis on Σc that seems reasonable to include is its

compact-ness. In fact if we consider on Cn _{the function}

H(p, q) := q1,

we get the following vector eld, whose orbits are open: XH = ∂p1.

Furthermore we would like to remain in the smooth category in order to use techniques coming from dierential geometry. Therefore we assume that cis a regular value for H. Then Σcis a smooth submanifold by the implicit

function theorem. On the contrary if c would be a critical value on the one hand we would have z0 ∈ Σc such that d(z0)H = 0. Then XH(z0) = 0 and

we would have the trivial solution z(t) ≡ z0. On the other hand the

comple-ment of critical points would be invariant under the ow and noncompact. So, as we have said above, we cannot expect the existence of periodic orbits in general.

The next step is to take a closer look to the relationship between ω and Σc. The nondegeneracy of ω implies that

R := (T Σc)ω

is a one-dimensional subbundle of T M. Since the dimension of Σc is odd,

the restriction ω0 _{of ω to T Σ}

c is degenerate. Thus its kernel must be R and

so R ⊂ T Σc. The importance of this bundle relies in the next result.

Proposition 1.3.1. If c is a regular value of an Hamiltonian function H and Σc and R are dened as above, then

XH ∈ R.

Therefore periodic orbits correspond to closed leaves of the distribution R, i.e. embeddings γ : S1_{→ Σ}

c such that ˙γ ∈ R.

Proof. Let v ∈ T Σc. Then dH(v) = 0 yields −ω(XH, v) = 0and so XH ∈ R.

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have an autointersection point, then the two tangent vectors are equal at the intersection since the system (1.1) is autonomous.

Conversely assume that γ is a closed leaf and regard γ as a 1-periodic function. ˙γ and XH are parallel and we assume that they point in the same

direction by changing the orientation of γ if necessary. Then exists a positive 1-periodic function f such that

f (t) ˙γ(t) = XH(γ(t)).

Then we consider the real function g dened by the following equations    dg ds(s) = f (g(s)) g(0) = 0 .

Since f is positiveand bounded, g is a dieomorphism dened on all R. Then dγ

ds(g(s)) = f (g(s)) ˙γ(g(s)) = XH(γ(g(s))).

Therefore γ(g(s)) is a periodic solution. Its period is the smallest positive value s0 such that g(s0) = 1.

The proposition shows that the existence problem can be formulated only in terms of the relative position between ω and Σc. However it has been

proved that we cannot solve the problem in the armative for a generic hy-pesurface (Σ, R) ⊂ (M, ω): see for example (25; 20). Weinstein's point of view is a compromise between the approach based upon Hamiltonian equa-tions and the one which relies exclusively on the distribution R. Its success is rooted in its connection with another important eld: contact geometry. Therefore we begin with some introductory denitions from the contact set-ting.

Denition 1.3.2. A contact form α on a manifold Σ is a nowhere vanishing 1-form on T Σ such that dα is a symplectic form on the subbundle ξ := ker α. Remark 1.3.3. The denition immediately implies that Σ is odd-dimensional and, since dα is symplectic on ξ,

T Σ = (T Σ)dα⊕ ξ.

We can choose a generator R of (T Σ)dα _{by requiring that α(R) = 1.}

R is uniquely determined by the conditions (

ιRdα = 0 ,

α(R) = 1 . R is called the Reeb vector eld of α.

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The denitions and propositions below lie the ground for the connection between symplectic and contact geometry.

Denition 1.3.4. Let Σ ⊂ (M, ω) be a hypersurface in a symplectic mani-fold. A vector eld Y dened in a neighbourhood of Σ and transverse to Σ is a Liouville vector eld for Σ if

L_Yω = ω.

(In what follows we shall abbreviate the transversality condition as Y t Σ.) Denition 1.3.5. Let (Σ, α) be a contact manifold and (M, ω) a symplectic manifold. We say that (Σ, α) is a contact submanifold of (M, ω), and we write (Σ, α) ⊂ (M, ω), if there exists an embedding j : Σ → M of Σ as a hypersurface in M such that

j∗ω = dα.

From this denition is clear that being a contact submanifold is invariant under symplectomorphism.

Proposition 1.3.6. If (Σ, α) ⊂ (M, ω), then R ∈ R.

Proof. Since j∗_{ω = dα}_{the conclusion follows from the very denitions of R}

and R.

Proposition 1.3.7. Let Σ ⊂ (M, ω) be a compact hypersurface. The follow-ing conditions are equivalent:

i) there exists a contact form α on Σ such that (Σ, α) ⊂ (M, ω), ii) Σ has a Liouville vector eld Y ,

iii) exists a contact form α on Σ, a neighbourhood U of Σ and a dieomor-phism Ψ: Σ × (−ε, ε) → U which is the identity on Σ, such that

Ψ∗ω = d(etα).

Moreover any of them implies that there is a neighbourhood U of Σ and a function H : U → R such that 0 is a regular value for H and

Σ = {H = 0}, XH|Σ= R .

Proof.

i) ⇒ ii) First we observe that an application of the generalized Poincaré lemma gives the equivalence between i) and the apparently stronger condition: i') there is a neighbourhood of Σ and a 1-form λ on it such that

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So we can dene Y from the equation ιYω = λ. Then

L_Yω = ιYdω + d(ιYω) = dλ = ω.

The transversality of Y can be seen as follows. Let R be the Reeb eld of j∗_λ_{. Then 1 = λ(R) = ω(Y, R). Unless Y /∈ T Σ, this leads}

to a contradiction because (T Σ)ω _{= R}_{. As a byproduct we nd the}

symplectic splitting

T M |Σ= Span(Y, R) ⊕ ξ,

since ιYω|ξ= λ|ξ = 0.

ii) ⇒ iii) We set λ:= ιYω. Then dλ = ω and LYλ = λ.

Let Φtbe the ow of Y . It is well dened for small t in a neighbourhood

U of Σ, since Σ is compact. We can construct the dieomorphism Ψ : Σ × (−ε0, ε0) → U

(x, t) 7→ Φt(x).

Let ρt be the ow of the coordinate vector eld ∂t on Σ × (−ε0, ε0).

Then Ψ carries ∂tupon Y and coniugates their ows.

Let π : Σ × (−ε0, ε0) → Σ be the projection on the rst factor and

jt: Σ → Σ × (−ε0, ε0) the embedding of Σ at height t, then jt= ρtj0.

Dene

α := Ψ∗λ.

Then α(∂t) = λ(Y ) = 0and so α(x,t) = π∗jt∗α(x,t).

Now compute d dt(j ∗ tα) = j ∗ 0 d dt(ρ ∗ tΨ ∗ λ) = j₀∗Ψ∗LYλ = j0∗Ψ ∗ λ = j₀∗α. Therefore j∗

tα = etj0∗α. Applying π∗ to this equation we nd at last

αx,t= et(j0π)∗α.

Taking the dierential on both sides yields the conclusion. iii) ⇒ i) It is enough to put λ = (Ψ−1₎∗_(et_{α). Then}

(Σ × 0, α) ⊂ (Σ × (−ε, ε), d(etα)) ⇒ (Σ, λ) ⊂ (M, ω). In order to nish the proof we have to exhibit the function H. Let π0 _{be the}

projection upon the second factor in Σ × (−, ). The function H such that H ◦ Ψ = π0

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has the desired property. Since Ψ is a symplectomorphism it carries Xπ0 to

XH and identies Σ × {0} with Σ embedded in M. Therefore it is enough

to show that Xπ0 = R:

ιRd(etα)|t=0= dt(R) − α(R)dt + ιRdα = −dt

Remark 1.3.8. Condition (ii) implies that being a contact submanifold is property which is resistant to C1 _{perturbation, as long as Y remains}

transverse to the hypersurface.

Condition (iii) is also interesting because gives a neighbourhood of Σ which is foliated by contact hypersurfaces dieomorphic to Σ. Moreover Reeb vector elds over two of such hypersurfaces are conjugated up to a constant factor and so they share the same dynamical properties. For instance if one of them has a closed characteristic so does the Reeb eld over any other leaf of the foliation.

We will now exhibit a relevant class of contact manifolds within the setting already described in Section 1.2.

Example 1.3.9 (Cotangent bundles). If the particle moves freely on (M, g), the only term in the Hamiltonian is the kinetic energy K. Then the zero-section is made by stationary point of the system whereas all the hypersur-faces {K = c, c > 0} are of contact-type. Indeed, the vertical vector eld Y (η) := ˆIη(η) is transverse to each nonzero level since

dηK(Y (η)) = d dt t=0K(η + tη) = kˆπ(η)(η, η) = 2K(η) (1.9)

(N.B. this identity can be seen as an application of Euler's theorem for ho-mogeneous function on vector bundles).

Finally K(η) > 0 provided η 6= 0. Using local coordinates we get the two equalities

(

ιYλ = 0 ,

ιYdλ = λ .

The latter is equivalent to LYdλ = dλ, remembering Cartan's formula.

Then one of the criteria in Proposition 1.3.7 is satised and so every nonzero energy level is a contact submanifold. Furthermore if Rc is the Reeb vector

eld at energy c and η ∈ {K = c}, we know that dλ(Y (η), Rc) = 1. In order

to nd the relation between XK and Rc is sucient to compute

dλ(Y (η), XK(η)) = dηK(Y (η)) = 2K(η) = 2c.

Then,

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We point out that not only all levels are of contact-type but also that the dynamics upon them is conjugated up to a constant positive factor. If we call Φt the ow of Y at time t, then

LYdλ = dλ ⇒ Φ∗t(dλ) = etdλ.

From this and the homogeneity of K we nd dλ(dηΦt(XK(η)), dηΦt(ξ))Φt(η)= Φ ∗ tdλ(XK(η), ξ) = etdλ(XK(η), ξ) = −etdηK(ξ) = −etdη(K ◦ Φ−1t ◦ Φt)(ξ) = −etdΦt(η)(K ◦ Φ−t)(dηΦt(ξ)) = −e−tdΦt(η)K(dηΦt(ξ)).

Therefore from the denition of XK we nally get

XK(Φt(η)) = etdηΦt(XK(η)).

Introduce now a non-zero magnetic eld σ and endow T∗_M _{with the}

sym-plectic structure ωσ as in Example 1.2.4. Then the zero-section is still made

by stationary point and Y is still transverse to the other energy levels, how-ever Y fails to satisfy the condition about the Lie derivative. In fact since Y is vertical, from (1.8) we have

LYωσ = dλ

and so Y is not a Liouville vector eld for ωσ. One attempt could be to nd

a vertical vector eld Z(η) = ˆIη(α(ˆπ(η)), where α : M → T∗M is a 1-form,

such that ₍

(Y + Z) t {K = c} , LZdλ = ˆπ∗σ .

(1.10) Mimicking the calculations (1.9), the rst condition can be rewritten as

k_π(η)_ˆ η, η + α_π(η)_ˆ ) 6= 0. Moreover the second equation (1.8) yields

LZdλ = ˆπ∗(dα) .

Using the injectivity of ˆπ∗ _{the couple of conditions (1.10) rewrites as}

(

kπ(η)ˆ η, η + απ(η)ˆ 6= 0 ,

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So the second condition forces us to reduce to the case of exact magnetic elds whereas the rst one tells us that the system is expected to behave dierently on dierent energy levels. In fact consider an energy level {K = c} such that exists a primitive α of σ such that

∀η ∈ {K = c}, K(η) > K(α_π(η)_ˆ ).

Then this hypothesis and the Cauchy-Schwarz inequality imply kˆπ(η) η, η + απ(η)ˆ = kπ(η)ˆ (η, η) + kπ(η)ˆ η, απ(η)ˆ

> 2K(η) − 2pK(η)

q

K(αˆπ(η)) > 0 .

The quantity that has a crucial role here is the Mañé critical value c0:

c0 = c0(k, σ) := inf α| dα=σ sup q∈M K(αq) . (1.12)

The analysis we have made so far for exact magnetic elds yields c > c0 ⇒ {K = c}is contact-type .

A detailed analysis about how the dynamics changes with the energy level can be found in the recent article by K. Cieliebak, U. Frauenfelder and G.P. Paternain (10).

The opposite situation, namely the case in which σ is symplectic, was studied by V. Ginzburg and E. Kerman (31) as well. They have studied the existence of periodic orbits on low energy levels, trying to generalize the so-called Weinstein-Moser conjecture (47; 35) to this class of twisted cotangent bundles.

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Chapter 2

The conjecture

Alan Weinstein proposed his famous conjecture for the rst time in 1979 (48) inspired by the recent work of P. Rabinowitz (40), who established the existence of periodic orbits when Σ is the boundary of star-shaped domains in Cn_{. This result deeply impressed matematicians involved in Hamiltonian}

systems, however Weinstein was not satised with the hypothesis of the theorem since it was not invariant under symplectomorphisms. His intuition was to recognize that the radial vector eld r∂rwas one of the main ingredient

of the proof and that the properties of r∂r, which were essential for the proof,

were actually symplectic (i.e. preserved by symplectomorphisms). r∂r is the

prototype of what we have called a Liouville vector eld and turns Σ into a contact hypersurface.

2.1 The statement

We are now in position to state precisely the

(Weinstein conjecture). Let (M, ω) be a symplectic manifold and Σ ⊂ M a compact hypersurface. If Σ is a contact submanifold of M then it carries a closed characteristic.

Remark 2.1.1.

i) The conjecture is still open today, although it is commonly believed to be true since it was proven in the armative in many particular cases. ii) The original formulation of the Weinstein conjecture included the

addi-tional assumption

H1(Σ, R) = 0.

However, subsequently the condition on the rst cohomology group was dropped since almost all the approaches to the proof tempted so far do not rely on it.

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The presence of the hypothesis on the vanishing of H1_{(Σ, R) in the}

early statement is due to the fact that it can be used as a substitutive requirement in some instances as we will say later when we discuss Liouville domains.

iii) The conjecture can be stated equivalently without any reference to the symplectic environment:

Any compact contact manifold (Σ, α) carries a closed characteristic. Indeed every contact manifold can be embedded in its symplectization:

(Σ × R, d(etα)).

iv) The conjecture becomes false if the contact hypothesis is removed with-out replacing it with something else. M.-R. Herman showed in (25) that exists a proper smooth function on Cn _{(n > 2), which has an energy}

level without closed trajectories. Later the counterexample was rened by Ginzburg and Gürel in (20) exhibiting a C2 _{function on C}2 _{with the}

same properties.

The conjecture with this degree of generality is still open. However, it was proven to be true for several classes of contact submanifolds. In the next section we shall give a brief account of some of the techniques used through the years.

2.2 Approaches to the proof

One of the main guideline has been to regard the conjecture exclusively as a problem in contact geometry. However since the problem is too general the starting point has been to x a class C of manifolds characterized by some properties (of topological nature, for instance) and accordingly a class of contact forms Λ on the elements of C. This method works well with three-dimensional manifolds where contact forms were intensively studied and classied (see Giroux (23) and Eliashberg (17)) and culminated in the full answer given by Taubes in 2007.

Beginning from the early Nineties the Weinstein conjecture has been proven in the aermative for the following cases:

Case 1. Hofer (27):

C_H0 =dim Σ = 3 , Λ_H0 =λ | ker λ is overtwisted .

Case 2. Hofer (27):

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Case 3. Abbas, Cieliebak and Hofer (1):

C_ACH =dim Σ = 3 , ΛACH =

λ ker λ is supported by a planar open book

.

Case 4. Taubes (45):

CT =dim Σ = 3 , ΛT =λ is an arbitrary contact form .

However recently improvements in higher dimensions were made too. The following two results generalize Case 1 and Case 2 respectively.

Case 5. Albers and Hofer (5):

CAH =dim Σ = 2n + 1 ,

ΛAH =λ| ker λ is Plastikstufe-overtwisted .

Case 6. Niederkrüger and Rechtman (36):

C_{N R} =dim Σ = 2n + 1 , ΛN R= λ ∃ N ,→ Σ | 0 6= [N ] ∈ Hn+1(Σ, F2),

N carries a Legendrian open book

.

The following scheme summarizes the implications which hold between the results listed above.

(H1) ⇐ (T ) ⇒ (ACH) ⇒ (H0)

⇑ ⇑

(N R) (AH)

For further insights the reader can consult Hofer (26) and Hutchings (30).

The other big guiding principle towards a proof of the conjecture is to investigate the presence of periodic orbits for a given Hamiltonian system as the energy level changes. The typical results that are available with this approach are the existence on {H = a} for almost all values a, with respect to the Lebesgue measure in R, or for a belonging to a dense subset of R. Theorems of the rst kind are called `almost existence theorems' whereas the others are called `nearby existence theorems'. These results rely on the denition of symplectic capacities. These are symplectic invariants dened axiomatically for symplectic manifolds in the following way.

Denition 2.2.1. A map c which associates to every symplectic manifold of xed dimension 2n a number in [0, +∞] is a symplectic capacity if satises the three properties:

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C1. Monotonicity: c(M, ω) ≤ c(M0_{, ω}0₎_,

if there is a symplectic embedding (M, ω) ,→ (M0_{, ω}0_).

C2. Conformality: c(M, sω) = |s|c(M, ω), ∀s ∈ (0, ∞). C3. Nontriviality: c(B(1), dλ) = π = c(Z(1), dλ), where

• dλis the standard contact structure on R2n_,

• B(1) ⊂ Cn _{is the open unit ball,}

• Z(1) ⊂ Cn _{is the open cylinder q}12

+ p12= 1 . The notion of capacity was introduced by Ekeland and Hofer in 1990 ((14; 15)). In the same year Hofer and Zehnder constructed an explicit capacity cHZ in (29), whose value depends essentially on the existence of

periodic solutions of certain Hamiltonian systems on M. Let (M, ω) be a symplectic manifold and denote by H(M, ω) the space of real functions H on M satisfying:

P1. there exist UH open, KH compact and a constant m(H) such that

UH ⊂ KH ⊂ (M \ ∂M ), H(UH) ≡ 0, H(M \ KH) ≡ m(H),

P2. ∀x ∈ M, 0 ≤ H(x) ≤ m(H).

Here m(H) can be interpreted as the oscillation of the function. Consider the subset Ha(M, ω) ⊂ H(M, ω) whose elements are called admissible and

characterized by the property that all the periodic solutions for the associated Hamiltonian system (1.1) are constant or have period strictly greater than 1. These Hamiltonians can be seen as the ones having periodic solutions with `bad' properties. In fact it is interesting to know when there are functions on the complement set H(M, ω)\Ha(M, ω), namely functions that have a

periodic solutions with small non zero period T , 0 < T ≤ 1. This information is provided by the Hofer-Zehnder capacity dened by

cHZ(M, ω) := sup Ha(M,ω) m(H). In fact if C ≥ 0, then cHZ(M, ω) ≤ C ⇐⇒ ∀ H ∈ H(M, ω), m(H) > C ⇒ H /∈ H_a(M, ω) . Therefore if cHZ is nite H has a fast periodic solution, provided its

oscilla-tion is big enough. The connecoscilla-tion with the Weinstein conjecture relies on the following

Theorem 2.2.2 (Nearby existence). Let Σ ⊂ (M, ω) be a compact hypersur-face and let Σ×(−ε0, ε0) ,→ M be an embedding onto an open neighbourhood

U of Σ, in other words we are choosing a tubular neighbourhood for Σ. Then cHZ(U, ω) < ∞ ⇒ for a.e. ε ∈ (−ε0, ε0), Σ×{ε} carries a periodic orbit.

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Remark 2.2.3. The preceding theorem tells us that the Weinstein con-jecture holds true when we make the further assumption that Σ has an open neighbourhood with nite capacity cHZ. Indeed, Remark 1.3.8

im-plies that the dynamics of the Reeb eld on Σ is conjugated, up to a time reparametrization, to the dynamics on Σ × {ε}, for every ε. Then, the ex-istence of a periodic orbit on Σ follows from the fact that, thanks to the theorem, there is a periodic orbits on some Σ × {ε0}. This line of reason

leads us to consider the larger class of stable hypersurfaces, which contains the contact ones.

Denition 2.2.4. An hypersurface Σ is called stable if there exists an embedding Σ×(−ε0, ε0) ,→ M such that characteristic bundle Rεon Σ×{ε}

is independent of ε.

Cieliebak and Mohnke in (11) show that stability is equivalent to the existence of a stabilizing 1-form α on Σ, such that

R ⊂ Σdα, α|R6= 0.

The discussion made so far proves that

Corollary 2.2.5. A compact stable hypersurface Σ with nite capacity cHZ

carries a closed characteristic.

Remark 2.2.6. Properties (C.1) and (C.3) implies that every bounded open set in an Euclidean space has nite capacity and so the conjecture is fully established for hypersurfaces in Cn_{. This result dates back to Viterbo, who}

however used variational arguments for the proof (46).

Remark 2.2.7. We have seen how the introduction of a special kind of capacity can be a useful tool for a solution of the conjecture. However the capacity is not unique and many deep results in symplectic geometry are enclosed within the properties (C.1)-(C.3): maybe rigidity phenomena for symplectomorphisms are the most important. They were investigated by Gromov (24) and Eliashberg (16) during the Seventies and the Eighties. Furthermore proving the existence of a capacity is in general a dicult task, which requires hard analitycal and variational tecnhiques. See (28) if you want to know more about this topic.

After this short survey (more on the state of art can be found in (19)), let us start with the proof of the Weinstein conjecture, which we have worked on. The main ingredient is the free period action functional which was used by Rabinowitz in his already mentioned proof of the conjecture (40). Recently this functional was rediscovered by Cieliebak and Frauenfelder (8) in order to dene a Morse-Bott homology for a class of symplectic mani-folds. They called this homological theory Rabinowitz-Floer Homology (the shorthand is RF H) and used it to nd obstructions to certain kind

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of embeddings or to prove the existence of closed characteristics. Moreover very soon it was clear that RF H could be applied equally well to solve several classical problems in symplectic geometry. Albers and Frauenfelder exploited it to solve Moser's problem about leafwise intersections (3). Pa-pers by Cieliebak, Frauenfelder and Oancea (9) and by Abbondandolo and Schwartz (2) developed explicit calculations for cotangent bundles nding rehlations with the well known symplectic (co-)homology. Finally Cieliebak, Frauenfelder and Paternain extended these results to more general manifolds (the so-called stable tame case) and combined them with the theory of Mañé critical values on twisteed cotangent bundles (10). For a survey about RF H and its applications the reader can see (4). The scheme of the proof that we are going to describe is inspired by these papers (see in particular (3) and Section 4.3 in (10)) and uses ideas from RF H, although is self-contained and does not require the transversality theory which is essential in the con-struction of RF H.

The rst step will be to state what are the additional assumptions we need. The actual line of reasoning will be developed in the subsequent chapters.

2.3 The additional hypotheses

We have highlighted in Remark 2.1.1.ii that every contact manifold (Σ, α) can be embedded as a contact submanifold in its symplectization

Σ×R, d(etα) .

However it would be nice if the ambient symplectic manifold for Σ could be chosen with some compactness property. The following denition goes in this direction and sets up a class of manifolds which are interesting for our purposes.

Denition 2.3.1. A compact exact symplectic manifold with boundary (V, λ) is called a Liouville domain, if (Σ := ∂V, α := λ|∂V) is a contact

submanifold.

Every Liouville domain carries a Liouville vector eld Y dened by the equation ιYdλ = λ. Then the contact condition implies that Y points

out-wards through Σ and its ow gives coordinates (x, t) ∈ Σ×(−ε, 0] on a collar of Σ. LYλ = λimplies that λ = etα in these coordinates.

Hence we can paste along the boundary an exterior piece Vext:= Σ×[0, +∞),

dene on it the 1-form λext:= etα and construct the completion ˆV of V ,

that is the exact symplectic manifold without boundary ( ˆV , ˆλ) := (V qYVext, λ qYλext).

Every (Σ×{t}, et_α)_{is contact and thus V is the monotone union of Liouville}

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so its ow is complete on ˆV and without critical points in the exterior. These properties characterizes the manifolds that are completions of Liouville domains, as we see in the next proposition which we state without proof. Denition 2.3.2. Let (M, λ) be an exact symplectic manifold. Then

if there exists an exhaustion of Liouville domains (Vk, λ|Vk), such that

Vk⊂ Vk+1, M =

[

k∈N

Vk,

then M is called an exact convex symplectic manifold,

if the ow of its Liouville eld Y is complete, then M is said to be complete;

if Y 6= 0 outside a compact set, then M has bounded topology. Proposition 2.3.3. An exact convex symplectic manifold is complete and has bounded topology if and only if it is the completion of some Liouville domain.

Example 2.3.4 (Stein manifolds). A Stein manifold (V, J, f) is a classical example of an exact convex manifold. We have seen in Example 1.1.6 that is exact with Liouville form λ:= −df ◦ J. Suppose that a is a regular value and consider the manifold with boundary

Va:= {f ≤ a.}

Then Va is a Liouville domain. This can be seen as follows. Let g be the

compatible Riemann metric dened by

g(u, v) = d(λ)(J u, v)

and compute the Hamiltonian vector eld Xf through its very denition:

−df (u) = df ◦ J (J u) = λ(J u) = dλ(Y, J u) = −dλ(J Y, u). So we get

Xf = −J Y, ∇f = Y,

where ∇f is the gradient of f with respect to g. Hence we nd that Y points outward through ∂Va as we wanted. Since the set of critical values

is negligible we nd that V is an exact convex manifold. Furthermore if all the critical points of f are cointaned in a single compact set we get also that V has bounded topology. The completeness can always be achieved after a suitable reparametrization f 7→ β ◦ f (see Biran and Cieliebak (7)).

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It is convenient to dene morphisms between exact convex symplectic manifolds that are not merely symplectomorphisms. In fact we shall require that the 1-forms can change only up to a summand that is the dierential of a compactly supported function.

Denition 2.3.5. Let ψ : (M, λ) → (M0_{, λ}0₎ _{be a map between two exact}

symplectic manifolds. ψ is called exact if there exists a compactly supported function h on M, such that

ψ∗λ0= λ + dh.

Remark 2.3.6. Since the support of h is assumed to be compact if an exact manifold M embeds through an exact map into an exact convex manifold than M is convex, too. As a result convexity is a property which is well-dened up to exact dieomorphisms.

The ideal candidate class for the ambient symplectic manifolds are com-pletions of Liouville manifolds since they are exact and they behave nicely at innity.

The former feature allows for the denition of the period-free action for loops on M and, during the proof, it will give apriori estimates for the rst deriva-tive for functions belonging to a specic moduli space M. The latter feature will be important in nding C0_{-bounds on the same set M.}

Remark 2.3.7. Every compact hypersurface Σ in an exact convex sym-plectic manifold M can be embedded in ˆVΣ,M the completion of a Liouville

manifold in such a way that the neighbourhoods of Σ (in M and in ˆVΣ,M)

are isomorphic. Indeed, it suces to choose V := Vkwith k suciently large.

So we can work in the larger class of exact convex symplectic manifold. Now that we have said what the ambient manifold looks like we have to impose some further condition on Σ. We actually ask for two kinds of properties. The former is needed to develop tools necessary for the proof, such as the dening Hamiltonian and the action-period equality. The latter is composed by the displaceability condition only. It reects a symplectic geometry relationship between Σ and M and in fact it is related to other symplectic quantities such as cHZ.

Restricted contact type submanifolds

As far as the rst kind of properties is concerned, we have found out in Proposition 1.3.7 that if a hypersurface Σ in a symplectic manifold (M, ω) is contact then there exists a neighbourhood U of Σ such that:

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there exists a proper function H : U → (−ε0, ε0)such that

Σ = {H = 0 }and R = XH.

The hypersurfaces we are looking for are those for which λ and H are globally dened so that the free period action functional can be calculated for loops with values in the whole M. In other words we can pick U = M above. Denition 2.3.8. An hypersurface Σ in an exact convex symplectic man-ifold (M, λ) is called of restricted contact type if there exists an exact embedding of a Liouville domain (V, λ0₎ _{in (M, λ), with Σ = ∂V .}

This is equivalent to saying that

i) Σ is bounding, i.e. M \Σ is made by two connected componets and one of them has compact closure. We call this one the interior of Σ, the other the exterior;

ii) there exists a compactly supported function h on M such that Σ, (λ + dh)|Σ

is of contact type.

So if Σ is restricted contact type the rst point tells us that the function H provided by Proposition 1.3.7 can be extended from a small neighbourhood of Σ to the whole M in such a way that

H is proper,

H < 0 on the interior, H > 0 on the exterior, dH is compactly supported.

One such function is called a dening Hamiltonian for Σ. In order to fulll this requirement take simply H : Σ×(−ε0, ε0) → (−ε0, ε0) that is the

projection on the second factor. Then extend smoothly on the complement of Σ×(−ε0, ε0), putting

H ≡ −ε0 in the interior and H ≡ ε0 in the exterior.

The point b) gives a globally dened 1-form ˆλ:= λ + dh which is contact on Σand which still makes M into an exact convex manifold. By the means of ˆ

λwe can dene the free period action functional A for a loop γ :=R/T Z→ M

of arbitrary period T as follows:

γ 7→ Z R_/_{T Z} γ∗ˆλ − Z R_/_{T Z} H ◦ γ dt. Then if, γ is a curve on Σ which satises ˙γ = XH(γ),

A(γ) = Z R/T Z ˆ λ_γ(t) ˙γ(t) dt − Z R/T Z 0 dt

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= Z R_/_{T Z} ˆ λγ(t) XH( ˙γ(t)) dt = Z R/T Z ˆ λ_γ(t) R( ˙γ(t)) dt = Z R/T Z 1 dt = T.

Hence we have got the action-period equality for closed orbits:

A(γ) = T . (2.1)

Remark 2.3.9. If (Σ, α)⊂(M, λ) is a contact submanifold then the follow-ing couple of homological conditions is sucient in order to guarantee that Σis of restricted contact type.

0=[Σ]∈H2n−1(M, R): this implies that Σ is bounding. In codimension

1 singular homology is the same as the cobordism category. So there exists a smooth compact 2n manifold N which realizes the homology of Σ to 0: in other words Σ = ∂N. The other component is simply M \N, which is unbounded.

H1

dR(Σ, R) = 0 (this is the condition Weinstein included in the original

statement of the conjecture). Condition i') in Proposition 1.3.7 yields a 1-form λ0 _{on a neighbourhood U of Σ such that}

dλ0 = ω = dλ (2.2)

and λ0 _{is contact on Σ. Then Equation (2.2) implies that}

d(λ0− λ) = dλ0− dλ = ω − ω = 0.

The vanishing of the rst de Rham cohomology group therefore yields a function h such that λ0 _{= λ + dh. Multiplying h by a function χ that}

is equal to 1 near Σ and compactly supported in U gives the function ˆ

h := χh which is dened on the whole M and compactly supported. Finally ˆλ:= λ + dˆh is the required 1-form.

Displaceability

An important subset of symplectomorphisms are those which can be writ-ten as time 1-maps of Hamiltonian ows. We are interested in having a large set available and so we allow for non-autonomous Hamiltonian functions, even though with a periodic dependance on the parameter.

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Denition 2.3.10. Let Φ : (M, ω) → (M, ω) be a symplectic dieomor-phism. Φ is called Hamiltonian if there exists a function H : [0, 1]×M → R such that:

a) there exists a compact set K of M, such that, for every t, Hthas support

in K;

b) if ΦH is the ow at time 1 of XH, then Φ = ΦH.

In the following discussion we will assume that H can be extended to a function H :R_/_Z_{× M → R, since every Hamiltonian dieomorphism arises}

from a periodic Hamiltonian. If H : [0, 1]×M → R is a generic function such that Φ = ΦH, then we can deneH(t, z) := h(t)H(t, z)b , where h: [0, 1] → R is a non-negative function, with support in (0, 1) and R1

0 hdt = 1. This last

condition implies ΦHb = ΦH = Φ. The condition on the support tells us that

b

H has a periodic extension.

We will denote by Hc(M ) the set of functions that satisfy a) and by

Ham(M, ω)the set of Hamiltonian dieomorphisms. Then b) gives a surjec-tive map

π : Hc(M ) → Ham(M, ω)

H 7→ Φ_H.

The ber upon a dieomorphism represents the possible ways to realize it as a periodic mechanical movement. The energy of such a movement can be dened using the associated Hamiltonian.

Denition 2.3.11. Let H ∈ Hc(M ) and dene the function osc(H) as

follows. osc(H) : R_/_Z _{→ [ 0, +∞)} t 7→ max z∈MHt(z) − minz∈MHt(z). Then dene, kHk := Z R/_Z osc(H) dt. (2.3)

k · kinduces a corresponding function on Ham(M, ω) through the map π: kΦk := inf

H∈Hc(M )

kHk

H ∈ π−1(Φ) . (2.4) So kΦk expresses the `minimum' amount of energy which makes the mechan-ical movement Ψ possible. We call this new function the Hofer's norm. We stress the fact that this is not a norm (since Ham(M, ω) is not a vector space). However kΦk represents the distance between the identity map and Φ when we endow Ham(M, ω) with a suitable distance, called the Hofer's metric. An account of the properties of this metric can be found in (28) as well as in the monograph by L. Polterovich (39).

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Denition 2.3.12. Let A be a subset of (M, ω). The displacement en-ergy of A is given by eω(A) := inf Φ∈Ham(M,ω) kΦk| Φ(A) ∩ A = ∅ . (2.5)

A is called displaceable if eω(A) < +∞, namely there exists Φ such that

Φ(A) ∩ A = ∅.

Remark 2.3.13. Here are some observations about the displacement energy. Since the Hamiltonian functions considered are compactly supported,

a displaceable set is bounded, i.e. contained in a compact subset. The displacement energy decreases under the action of symplectic

em-beddings. Suppose Ψ : (M, ω) ,→ (M0_{, ω}0₎ _{is one such embedding and}

Φis in Ham(M, ω). Then Ψ ◦ Φ ◦ Ψ−1 dened on the image of Ψ can be extended to an element Φ0 _{of Ham(M}0_{, ω}0₎ _{simply imposing}

Φ0(z) = z, z /∈ Ψ(M ).

This new element satises kΦ0_{k ≤ kΦk}_{because we have also an}

exten-sion map Hc(M ) → Hc(M0) which maps H to an H0 dened in the

obvious way. Then kHk = kH0_k _{and the commutativity relation}

π0(H0) = (π(H))0

yields kΦ0_{k ≤ kΦk}_{. Furthermore if Φ displaces A, then Φ}0 _displaces

Ψ(A)and so

eω(A) ≥ eω0(Ψ(A)).

In a xed symplectic manifold (M, ω) the displacement energy is mono-tone:

A ⊂ B ⇒ e_ω(A) ≤ eω(B).

The Hofer's norm and, hence, the displacement energy are positively homogeneous with respect to the symplectic form:

∀a > 0, e_aω= |a|eω.

The displacement energy is outer regular. Namely if eω(A) < +∞and

ε > 0is xed, then there exists a neighbourhood Uε of A such that

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As we have mentioned few pages ago the displacement energy is tied to another important geometric quantity, namely the Hofer-Zehnder ca-pacity. This is done via the energy-capacity inequality. Several results of this kind are obtained under distinct assumptions. F. Schlenk stud-ied this problem in (43). One of the corollaries he gets is the following one.

Theorem 2.3.14. Let (M, ω) be a symplectic manifold geometrically bounded (M the completion of a Liouville domain is sucient). If A is a subset of M, then

cHZ(A) ≤ 4eω(A).

Example 2.3.15 (Bounded sets in linear spaces). Every bounded set B in Cn is easily seen to be displaceable. Any translation by a vector v where v is of the form v = P_kvk∂_qk is in Ham(Cn, dλ). It is enough to take

H(p, q) =X

k

vkqk.

Call Φt the ow of XH. In order to nd a compactly supported function,

whose ow at time 1 displaces B, simply multiply H by a cut-o function which is constantly equal to 1 in a neighbourhood of the bounded set

[

t∈[0,1]

Φt(B).

2.4 The main theorem

We are now ready to state the theorem we are going to prove in the subsequent chapters.

Theorem 2.4.1. Let (M, λ) be an exact convex symplectic manifold and let Σ be a compact hypersurface contained in M. If Σ is restricted contact type and displaceable then it carries a contractible closed characteristic whose period is smaller than edλ(Σ).

The manifolds which best suit the hypotheses of the theorem are subcrit-ical Stein manifolds. For a generic Stein manifold (V, J, f) it is possible to choose f as a Morse function whose critical points have index less or equal to half the dimension of V . If the inequality is strict, then V is called sub-critical. These manifolds has been studied by Biran and Cieliebak (7), who discovered that every compact subset is displaceable.

Remark 2.4.2. In Remark 2.3.13 we have mentioned the energy-capacity inequality. This inequality allows for a comparison between the theorem

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presented here and the theorem of nearby existence developed by Hofer and Zehnder, which is easily seen to be stronger. Indeed,

• eω(Σ) < +∞ ⇒ cHZ(Σ) < +∞,

• Σ restricted contact type ⇒ Σ stable submanifold.

Therefore the hypotheses of Theorem 2.4.1 implies those of Corollary 2.2.5, which was a consequence of the Nearby Existence Theorem 2.2.2. On the other hand, recently Cieliebak, Frauenfelder and Paternain have succeeded in extending the denition of RF H to the larger class of stable tame manifolds. As a byproduct they improved Theorem 2.4.1 substituting the restricted contact type hypothesis with the slightly relaxed stable tame hypothesis. However the gap between the energy-capacity inequality methods and those based on the free period action functional is still wide and it is likely to remain so. We have decided to not present the theorem in this strong and up-to-date version because new ideas come into play in its proof that are not merely a generalization of the simple case.

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Chapter 3

The free period action

functional

We have described in the rst chapter how Newton's physics can be encoded in the language of Hamiltonian systems. The latter formulation presents some advantages respect to an approach merely based on the Sec-ond Law of Dynamics: there is a group of transformation which preserves the dynamics (symplectic dieomorphism) and many stability results are known. But perhaps the most appealing feature is the possibility to get Hamilton equations via a variational argument. The `admissable' or phys-ical motions are characterized by the fact that they are critphys-ical points of a suitable functional dened on a space of smooth paths in the conguration space. However, since the domain of the functional is innite-dimensional, establishing the existence of critical points is quite a dicult task. Several properties were singled out which are sucient for a functional in order to have critical points (the most important are probably the direct method and the minimax method), but unfortunately these do not apply directly to the action functional of classical mechanics on the space of loops. The major diculty is that the critical points of the action do not have nite Morse index. Rabinowitz was the rst in 1978 (40) to circumvent the problem and to exploit variational properties of the action. However it was only with the work of A. Floer that a general theory has been available. Floer in (18) con-structed an homology theory, whose complex is generated by critical points. Therefore if we can compute the homology, we will gain information also about the critical points. Although we will not construct an homology the-ory for the action à la Floer, the proof will share some basic lemmas with Floer's theory. In this rst chapter we will dene a family of free period action functionals, see that they admit a gradient-like system and establish some properties of the solutions with nite energy.

An approach to the Weinstein conjecture via J-holomorphic curves